This will be a course about Coxeter groups, their Hecke algebras, Kazhdan-Lusztig polynomials and Soergel bimodules. It will culminate in an algebraic proof of the Kazhdan-Lusztig conjecture (a characterformula for highest weight modules over a complex semi-simple Lie algebra, which up until recently had difficult geometric proofs) and a proof of the positivity of Kazhdan-Lusztig polynomials (a conjecture which has been open for over thirty years). Along the way ideas from categorification, higher algebra and Hodge theory play an essential role.